M-M-1 queue

The queue length in an $M/M/1$ queue is a birth-death process with constant arrival rate $\lambda$ and service rate $\mu$ while nonempty.

If potential arrivals form a Poisson process of rate $\lambda$ and an arrival seeing $n$ customers joins with probability $p(n)$, the queue length is a birth-death process with birth rate $b_n=\lambda p(n)$ and death rate $d_n=\mu$ for $n\ge 1$. Its stationary ratios, when normalizable, are

An $M/M/1$ queue is positive recurrent exactly when its traffic intensity $\rho =\lambda /\mu$ is below one.

When $\rho <1$, the equilibrium queue length has distribution ${\pi }_n=(1-\rho ){\rho }^n$ and mean $\rho /(1-\rho )$.

At equilibrium, the departure process of a stable $M/M/1$ queue is Poisson with the same rate as the arrival process.

If a served customer returns with probability $p$, thinning service completions gives effective external departure rate $\mu (1-p)$ for the queue-length process.

In equilibrium, a Bernoulli-feedback $M/M/1$ queue has the same queue-length process as an ordinary queue with service rate $\mu (1-p)$, so its external departures form a Poisson process of rate $\lambda$.

Starting with one customer, the mean busy period of an $M/M/1$ queue iswhen $\lambda <\mu$. It is infinite when $\lambda \ge \mu$.